Write N(Cop)N(C^{op}) for the ordinary nerve of the ordinary category CopC^{op} (passing to the opposite category is just a convention here, with no effect on the substance of the statement). Then an ∞\infty-pseudofunctor or (∞,1)-presheaf or homotopy presheaf on CC is a morphism of simplicial sets

F:N(Cop)→N(KanCplx).
F : N(C^{op}) \to N(\mathbf{KanCplx})
\,.

One sees easily in low degrees that this does look like the a pseudofunctor there:

the 1-cells of N(C)N(C) are just the morphisms in CC, so that on 1-cells we have that FF is an assignment

which means that FF does not necessarily respect the composition of moprhisms, but instead does introduce homotopiesF(f,g)F(f,g) for very pairs of composable morphisms, which measure how F(g)∘F(f)F(g)\circ F(f) differs from F(g∘f)F(g \circ f). These are precisely the homotopies that one sees also in an ordinary pseudofunctor. But for our (∞,1)(\infty,1)-functor there are now also higher and higher homotopies:

the 3-cells of N(C)N(C) are triples of composable morphisms (f,g,h)(f,g,h) in CC. They are sent by FF to a tetrahedron that consists of a homotopy-of-homotopies from the F(f,g)⋅F(h,g∘f)F(f,g) \cdot F( h , g\circ f ) to F(g,h)⋅F(f,h∘g)F(g, h) \cdot F(f , h \circ g);

Then: every (∞,1)(\infty,1)-functor N(Cop)→∞GrpdN(\mathbf{C}^{op}) \to \infty Grpd is equivalent to a strictly composition respecting functor of this sort. Precisely: write [Cop,KanCplx]∘[\mathbf{C}^{op}, \mathbf{KanCplx}]^\circ for the full sSetsSet-enriched subcategory on those strict functors that are fibrant and cofibrant in the model structure on simplicial presheaves on C\mathbf{C}. Then we have an equivalence of ∞-groupoids